Geometric evolution of complex networks

Abstract

We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time $t$ are distributed homogeneously between a new node and the exising nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature $\beta$ and general time-dependent chemical potential $\mu (t)$. The chemical potential limits the spatial extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that $\mu (t)$ can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network---its $\text{history}$---the degree-degree correlation can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient $⟨c⟩$. In the thermodynamic limit, we identify a phase transition between a random regime where $⟨c⟩\to 0$ when $\beta <{\beta }_{\mathrm{c}}$ and a geometric regime where $⟨c⟩>0$ when $\beta >{\beta }_{\mathrm{c}}$.

References in corpus (8)

• Hyperbolic Geometry of Complex Networks
• Navigability of Complex Networks
• Self-similarity of complex networks and hidden metric spaces
• Curvature and temperature of complex networks
• Emergence of Soft Communities from Geometric Preferential Attachment
• Clustering implies geometry in networks
• Dimensionality of social networks using motifs and eigenvalues
• Continuum percolation with steps in an annulus